# Properties

 Label 3.3.af_p_abg Base Field $\F_{3}$ Dimension $3$ Ordinary Yes $p$-rank $3$ Principally polarizable Yes Contains a Jacobian No

## Invariants

 Base field: $\F_{3}$ Dimension: $3$ L-polynomial: $( 1 - x + 3 x^{2} )( 1 - 4 x + 8 x^{2} - 12 x^{3} + 9 x^{4} )$ Frobenius angles: $\pm0.0540867239847$, $\pm0.406785250661$, $\pm0.445913276015$ Angle rank: $2$ (numerical) Jacobians: 0

This isogeny class is not simple.

## Newton polygon

This isogeny class is ordinary. $p$-rank: $3$ Slopes: $[0, 0, 0, 1, 1, 1]$

## Point counts

This isogeny class is principally polarizable, but does not contain a Jacobian.

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 6 1020 22536 346800 10497066 383112000 11123580558 284453683200 7461772521912 204245087135100

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -1 15 32 47 169 720 2323 6607 19256 58575

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ab $\times$ 2.3.ae_i and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
Endomorphism algebra over $\overline{\F}_{3}$
 The base change of $A$ to $\F_{3^{4}}$ is 1.81.ao 2 $\times$ 1.81.ah. The endomorphism algebra for each factor is: 1.81.ao 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-2})$$$)$ 1.81.ah : $$\Q(\sqrt{-11})$$.
All geometric endomorphisms are defined over $\F_{3^{4}}$.
Remainder of endomorphism lattice by field
• Endomorphism algebra over $\F_{3^{2}}$  The base change of $A$ to $\F_{3^{2}}$ is 1.9.f $\times$ 2.9.a_ao. The endomorphism algebra for each factor is:

## Base change

This is a primitive isogeny class.

## Twists

Below are some of the twists of this isogeny class.
 Twist Extension Degree Common base change 3.3.ad_h_aq $2$ 3.9.f_af_acs 3.3.d_h_q $2$ 3.9.f_af_acs 3.3.f_p_bg $2$ 3.9.f_af_acs
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 3.3.ad_h_aq $2$ 3.9.f_af_acs 3.3.d_h_q $2$ 3.9.f_af_acs 3.3.f_p_bg $2$ 3.9.f_af_acs 3.3.ad_h_aq $4$ (not in LMFDB) 3.3.af_r_abi $8$ (not in LMFDB) 3.3.ad_j_ao $8$ (not in LMFDB) 3.3.ab_b_c $8$ (not in LMFDB) 3.3.ab_f_ac $8$ (not in LMFDB) 3.3.b_b_ac $8$ (not in LMFDB) 3.3.b_f_c $8$ (not in LMFDB) 3.3.d_j_o $8$ (not in LMFDB) 3.3.f_r_bi $8$ (not in LMFDB) 3.3.ad_g_an $24$ (not in LMFDB) 3.3.ab_c_al $24$ (not in LMFDB) 3.3.b_c_l $24$ (not in LMFDB) 3.3.d_g_n $24$ (not in LMFDB)